Pullback and Pushout Constructions in C*-Algebra Theory

A systematic study of pullback and pushout diagrams is conducted in order to understand restricted direct sums and amalgamated free products of C*-algebras. Particular emphasis is given to the relations with tensor products (both with the minimal and the maximal C*-tensor norm). Thus it is shown that pullback and pushout diagrams are stable under tensoring with a fixed algebra and stable under crossed products with a fixed group. General tensor products between diagrams are also investigated. The relations between the theory of extensions and pullback and pushout diagrams are explored in some detail. The crowning result is that if three short exact sequences of C*-algebras are given, with appropriate morphisms between the sequences allowing for pullback or pushout constructions at the levels of ideals, algebras and quotients, then the three new C*-algebras will again form a short exact sequence under some mild extra conditions. As a generalization of a theorem of T. A. Loring it is shown that each morphism between a pair of C*-algebras, combined with its extension to the stabilized algebras, gives rise to a pushout diagram. This result has applications to corona extendibility and conditional projectivity. Finally the pullback and pushout constructions are applied to the class of noncommutative CW complexes defined by (S. Eilers, T. A Loring, and G. K. Pedersen J. Reine Angew. Math., 1998, 499, 101–143) to show that this category is stable under tensor products and under restricted direct sums.